


import math
def 				_UpperCAmelCase							(UpperCamelCase__	:    int		):

       if 1 < number < 4:
              # 2 and 3 are primes
              return True
       elif number < 2 or number % 2 == 0 or number % 3 == 0:
              # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
              return False

       # All primes number are in format of 6k +/- 1
       for i in range(5							,				int(math.sqrt(UpperCamelCase__		) + 1		)							,				6		):
              if number % i == 0 or number % (i + 2) == 0:
                     return False
       return True
def 				_UpperCAmelCase							(UpperCamelCase__	:    List[Any] = 0.1		):
       _A :			Any							       =					3
       _A :			str							       =					3

       while primes / (2 * j - 1) >= ratio:
              for i in range(j * j + j + 1							,				(j + 2) * (j + 2)							,				j + 1		):
                     primes += is_prime(UpperCamelCase__		)
              j += 2
       return j


if __name__ == "__main__":
    import doctest

    doctest.testmod()

'''simple docstring'''




import numpy as np






def __lowerCAmelCase					(				UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							)      ->       str:
    __lowerCamelCase            =    int(np.ceil((x_end - xa) / h							)							)
    __lowerCamelCase            =    np.zeros((n + 1,)							)
    __lowerCamelCase            =    ya
    __lowerCamelCase            =    xa

    for k in range(UpperCamelCase__							):
        __lowerCamelCase            =    f(UpperCamelCase__							,    y[k]							)
        __lowerCamelCase            =    f(x + 0.5 * h							,    y[k] + 0.5 * h * ka							)
        __lowerCamelCase            =    f(x + 0.5 * h							,    y[k] + 0.5 * h * ka							)
        __lowerCamelCase            =    f(x + h							,    y[k] + h * ka							)
        __lowerCamelCase            =    y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
        x += h

    return y


if __name__ == "__main__":
  import doctest

  doctest.testmod()
